Hogg. — On the Inscribed Parabola. 493 



Solving for mn, nl, Im from the equations 



— Smna^ + 7db'^ + Imc^ = 2k ax 

 mna^ — Snlb''^ + Imc^ = 2k by 

 mna^ -\- nlb^ — 3Zmc'^ = 2k cz, 

 we obtain 



a^mn : b^nl : cHm = 2ax + by + cz : ax -{- 2by + cz : ax -\- by -\- 2cz ; 

 hence the locus of the point of contact Q is 



2ax + by -\- cz ax -\- 2by -\- cz ax -\- by + 2cz 

 which on expansion becomes 



abc {ayz + bzx + cxtj) + {ax + by -\- cz) x 

 [{a;' + 262 _^ 2c2) ax + (6'^ + 2o^ + 2a-) 6?/ + (c- + 2a'' + 26^) c^] = o, 

 a circle of radius 4R, having its centre at the point whose trilinear ratios 

 are 



[cos A — cos B cos C, cos B — cos C cos A, cos C — cos A cos B] . 



The triangle formed by the lines 



2ax + by -\- cz = o, ax -{- 2by + cz = o, ax + by + 2cz = o 

 may be called the Aq. 



4. Let S = Vlax + Vmby + Vncz = o 

 S' = Vl'ax + Vm'by + Vn'cz = o, 



when r = m — n, m' = n — I, n' = I — vi, be two parabolas inscribed in 

 the triangle ABC. 



Theequationof the line joining their foci F (-7, -, -) andF' ( y^,, — ,, ,1 



\ i 711/ IbJ \i 776 /v / 



,„ a? , y , , z 



is a - -i- mm f + 7in - — o, 



a b c 



showing that FF' is a chord of the circle ABC passing through the 



symmedian point (a, b, c) of the triangle ABC. 



The centres of perspective P, P' of the triangle ABC and the two 

 triangles formed by joining the points of contact of S and S' with the 

 sides of the triangle ABC have trilinear ratios 



\la' mb' ncj' \l'a' m'b' n'cj 

 respectively : the equation of the line PP' is Wax + mm'by + nn'cz = o, 



which is satisfied by the trilinear ratios of the centroid ( -, j, -j, showing 



that P and P' are the extremities of a diameter of the Steiner ellipse of 

 the triangle ABC. Hence the lines joining the fourth point of inter- 

 section of the circle ABC and the Steiner ellipse of the triangle to the 

 extremities of any chord of the circle, which passes through the 

 symmedian point of the triangle, meet the Steiner ellipse again at the 

 extremities of a diameter of the ellipse. 



The polars of the centroid of the triangle ABC with respect to S and 

 S' form a pair of parallel tangents to the Steiner ellipse of the triangle 

 ABC. The equations of the two polars are Pax + m%y + n'cz = o, 

 I'^ax + m'-by + n''^cz = o, and at their intersection ax : by : cz = It' : vim' : nn', 

 showing that the tangents meet on the line at infinity. 



